Unit 5 Curve Sketching and Optimization: Using Derivatives to Understand and Optimize Functions

Function graph (f)

A graph that shows the output values of a function f(x) (the “heights” of the function) across x-values.

First derivative (f′)

A function whose value f′(x) equals the slope of the tangent line to f at x; it describes how fast f is changing and whether f is increasing or decreasing.

Second derivative (f″)

A function whose value f″(x) describes how the slope f′(x) is changing; it is used to determine the concavity of f.

Increasing (via derivative)

f is increasing on an interval where f′(x) > 0 (tangent slopes upward left-to-right).

Decreasing (via derivative)

f is decreasing on an interval where f′(x) < 0 (tangent slopes downward left-to-right).

Horizontal tangent

A point on f where the tangent line has slope 0; this occurs where f′(x) = 0.

Critical point (critical number)

An x-value c where f′(c)=0 or f′(c) does not exist (while f is defined); it is a candidate location for local extrema.

Local maximum (from f′)

A point where f changes from increasing to decreasing; typically where f′ changes sign from positive to negative at x=c (with f′(c)=0 or undefined).

Local minimum (from f′)

A point where f changes from decreasing to increasing; typically where f′ changes sign from negative to positive at x=c (with f′(c)=0 or undefined).

First Derivative Test

A method that classifies a critical point by checking sign changes of f′: +→− gives a local max; −→+ gives a local min; no sign change means no local extremum.

Concave up

A shape where slopes are increasing; occurs on intervals where f″(x) > 0.

Concave down

A shape where slopes are decreasing; occurs on intervals where f″(x) < 0.

Inflection point

A point where f changes concavity (concave up ↔ concave down); typically where f″(c)=0 or DNE AND f″ changes sign around c.

Sign change requirement (for extrema/inflection)

A reminder that f′(c)=0 alone does not guarantee an extremum, and f″(c)=0 alone does not guarantee an inflection point; a sign change (or corresponding behavior change) must be verified.

Local extremum of f′

A point where f′ has a local maximum or minimum; often occurs where f″(x)=0 with a sign change in f″.

Steepest slope (on f)

The x-value where f has its largest slope; equivalently where f′ is maximized (so f′ has a local maximum and typically f″=0 with a sign change).

Piecewise-linear function

A function made of line segments; on each segment the slope is constant, so f′ is constant on each corresponding interval.

Corner/cusp (nondifferentiable point)

A sharp point on a graph where the derivative typically does not exist; in f′ this appears as a break or an undefined value at that x.

Domain (in curve sketching/optimization)

The set of x-values for which f is defined; it determines the intervals used for sign charts and the feasible set of values in word problems.

Objective function

The function representing the quantity to maximize or minimize (e.g., area A(x), surface area S(x), cost C(x), distance D(x)).

Constraint equation

An equation relating variables (e.g., fixed perimeter, fixed volume) used to rewrite the objective function in terms of a single variable.

Closed Interval Method

A method for absolute extrema on a closed interval: find critical numbers in the interval, evaluate the function there and at endpoints, then compare values to identify absolute max/min.

Second Derivative Test

A shortcut at a critical point c where f′(c)=0: if f″(c)>0 then local min; if f″(c)<0 then local max; if f″(c)=0 then inconclusive.

Distance-squared trick

In distance minimization, minimizing D is equivalent to minimizing D² (since √ is increasing), which often simplifies algebra by removing square roots.

Feasible domain (optimization)

The physically possible values for the variable in a word problem (e.g., lengths > 0); used to reject invalid critical points and to ensure endpoints are checked when appropriate.